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An approach based on shortest path and connectivity consistency for sensor network localization problems

Makoto Yamashita (Tokyo Institute of Technology)I-Lin Wang (National Cheng Kung University)Zih-Cin Lin (National Cheng Kung University)

2012/08/22 ISMP 2012 (TU Berlin, Berlin, Germerny)

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Outline

Sensor Network Localization1• Mathematical Formulation

Framework of our approach2• Shortest Path• Gradient Method• Connectivity Consistency

Numerical Results3Multiple Start4

• Starting Point Selection• Combination of Location Results

Conclusion and Future works5

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SNL(Sensor Network Localization Problem)

We want to infer locationsfrom distance information

System of Equation

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4

Protein Structure

We can use distances between atoms measured by NOE effect.

We want to infer whole structure.

Structure determines chemical property of protein.

There are many other applications.

2012/08/221AX8, 1003 atoms

5

Existing Methods

Multidimensional Scaling[Merit] Low computation cost[Demerit] All distances are necessary

SDP relaxation (Biswas & Ye 2004)[Merit] High accuracy[Demerit] High compuation cost

We combine some heuristicsMiddle accuracy & Middle computation cost

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Our Approach

Combination of heuristicsShortest pathGradient methodConnectivity consistency

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Trilateration

Three anchors determine the location uniquely.

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1a

2a

3a

1d

2d

3d

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Shortest Path

Propagation from anchors

Moredistance information⇒Shortest Path

Rough estimate Gradient method⇒

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Minimization of difference

Instead of solving the system

minimize

Effective for noisy distance input

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input true noise(e.g.:20% ～ 30%)

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Gradient Method

Repeat

until

2012/08/22Shortest Path result

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Connectivity Consistency

Given distance is usually less than radio range.

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Repulsion

Attraction

Adjustment

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Framework of our heuristics

1. Select initial anchors2. Estimate roughly with Shortest Path3. Apply Gradient Method with estimate distance4. Apply Gradient Method with original distance5. Adjust sensors by Connectivity Consistency6. Go to Step 4

until there is no significant improvement

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Numerical Experiments

Effect of Shortest Path & Consistency AdjustmentSNLsa (Shortest Path & Consistency Adjustment)

vs. SNLa (Consistency Adjustment)vs. SNLs (Shortest Path)

Comparison with SDP relaxationSNLsa vs. SFSDP (Kim et at, 2009)

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Sparse Full SDP relaxation

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Test Instances & RMSD

[0,1]x[0,1] space in 2D exact distance (zero noise) #sensors = 200, 500, 1000 #anchors = #sensors/10 radiorange = 0.05, 0.10, 0.15, 0.20, 0.25, 0.30 average of 100 randomly generated instances

Evaluate RMSD (Root Mean Square Deviation)

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SNLsa vs. SNLa vs. SNLs

The accuracy of SNLa is poor⇒Shortest Path is effective

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0.3 0.25 0.2 0.15 0.1 0.051.00E-071.00E-061.00E-051.00E-041.00E-031.00E-021.00E-01

1.00E+001.00E+01

0

5

10

15

20

25

et_SNLsaet_SNLaet_SNLsrmsd_SNLsarmsd_SNLarmsd_SNLs

radiorange

RMSD

time(s)

0.3 0.25 0.2 0.15 0.1 0.051.00E-071.00E-061.00E-051.00E-041.00E-031.00E-021.00E-01

1.00E+001.00E+01

0

2

4

6

8

10

12

14

et_SNLsaet_SNLaet_SNLsrmsd_SNLsarmsd_SNLarmsd_SNLs

radiorange

RMSD

time(s)

1000sensors

500sensors

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SNLsa vs. SNLs vs. SNLa (2)

For middle radioranges, Consistency Adjustment works well.

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0.3 0.25 0.2 0.15 0.1 0.051.00E-07

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+01

0

0.5

1

1.5

2

2.5

3

3.5

et_SNLsaet_SNLaet_SNLsrmsd_SNLsarmsd_SNLarmsd_SNLs

radiorange

RMSD

time(s)

200 sensors

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SNLsa vs. SFSDP

For large radioranges, SNLsa is faster.

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0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.11.00E-07

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+01

0.00

2.00

4.00

6.00

8.00

10.00

12.00

et_SFSDPet_SNLsarmsd_SFSDPrmsd_SNLsa

radiorange

RMSD

time(s)

0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.11.00E-07

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+01

0.00

2.00

4.00

6.00

8.00

10.00

12.00

et_SFSDPet_SNLsarmsd_SFSDPrmsd_SNLsa

radiorange

RMSD

time(s)

1000sensors

500sensors

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Multiple Start

For the starting anchors, there are many candidates.

We list-up cliques of size 4 and select better cliques.(e.g.: large volume of triangle or tetrahedra.)

Each starting anchors generates different locations.

We want reasonable result from multiple solutions.

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Different Solutions

For similar solutions, we can take their average.

Lack of edges often make the instance harder.

We need different approach to select locations.

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Densest Subset

1. Collect all solutions for each sensor.

2. Generate a graph by connecting each other.

3. Find the densest subsetvia discrete optimization.(Nagano et al, 2011)

4. Take the average ofthe densest subset.

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Numerical Results of Densest Subset Protein 1HOE (2D projection)

Protein 1KDH (2D projection)

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Clique 1 2 3 4 5 6

RMSD 0.223434 0.223498 0.223498 0.223498 0.223498 0.223498

7 8 9 10 11 12

0.223498 1.416665 4.620514 6.158431 6.189689 7.732091

RMSD(Densest Average) = 0.227412

Clique 1 2 3 4 5 6

RMSD 1.356912 1.356923 1.356971 1.356971 1.356971 1.356971

7 8 9 10 11 12

1.356971 1.356971 1.356971 4.178180 10.286665 12.354961

RMSD(Densest Average) = 1.248964

Ignoring deviations, we obtain reasonable solution.

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Conclusion and Future Works

Shortest Path & Consistency Adjustment works well for randomly generated instances.

Combination of multiple starts generatesreasonable solutions.

We should discuss multiple sensor types.We should introduce chemical property of proteins.

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謝謝聆聽 , Thank you very much for your attention.